# Tagged Questions

**1**

vote

**0**answers

118 views

### Dalzel's integral for $\pi$ and the lemniscate constant

$\pi=\varpi_2$ can be considered as a member of the sequence of real numbers
$$\varpi_m=2\int\limits_0^1\frac{dt}{\sqrt{1-t^m}},$$
and, for example, the Wallis product formula ...

**76**

votes

**5**answers

6k views

### Source and context of $\frac{22}{7} - \pi = \int_0^1 (x-x^2)^4 dx/(1+x^2)$?

Possibly the most striking proof of Archimedes's inequality $\pi < 22/7$ is an integral formula for the difference:
$$
\frac{22}{7} - \pi = \int_0^1 (x-x^2)^4 \frac{dx}{1+x^2},
$$
where the ...

**1**

vote

**1**answer

303 views

### Best rational approximation in a special sense

Let $\alpha$ be an irrational number, $n\geq 1$ and
$ X_n=\lbrace (x,y) \in {\mathbb Z}^2 | |y| \leq n, \ x+y\alpha >0 \rbrace$
Now let $(x_n,y_n)$ minimize the quantity $x+y\alpha$ on $X_n$. ...

**7**

votes

**3**answers

1k views

### Kronecker Approximation theorem and Fibonacci numbers

There is a famous old theorem by Kronecker that for every positive real $\alpha$ and $\epsilon>0$ there exists a positive integer n such that $\alpha n$ is within $\epsilon$ of an integer.
...

**26**

votes

**1**answer

1k views

### An inequality for cosine of n

Can anyone provide a proof of the following inequality?
If $n$ is a positive integer, $n\geq2$, then $$\cos(n) \leq 1 - 2^{-n}.$$
This is satisfied if $n$ is not within about $2^{-n/2}$ of a multiple ...

**19**

votes

**3**answers

2k views

### When is $n/\ln(n)$ close to an integer?

As usual I expect to be critisised for "duplicating"
this question. But I do not! As Gjergji immediately
notified, that question was from numerology. The one I ask you here
(after putting it in my ...